Origins and optimization of entanglement in plasmonically coupled quantum dots

A system of two or more quantum dots interacting with a dissipative plasmonic nanostructure is investigated in detail by using a cavity quantum electrodynamics approach with a model Hamiltonian. We focus on determining and understanding system configurations that generate multiple bipartite quantum entanglements between the occupation states of the quantum dots. These configurations include allowing for the quantum dots to be asymmetrically coupled to the plasmonic system. Analytical solution of a simplified limit for an arbitrary number of quantum dots and numerical simulations and optimization for the two- and three-dot cases are used to develop guidelines for maximizing the bipartite entanglements. For any number of quantum dots, we show that through simple starting states and parameter guidelines, one quantum dot can be made to share a strong amount of bipartite entanglement with all other quantum dots in the system, while entangling all other pairs to a lesser degree.

Figure 1

Time dependence of the populations of the states $|S;s=0\rangle , |A;s=0\rangle$, and $|q_2=0,q_1=0;s=1\rangle$, and the concurrence, for a two-QD system initially in the $|q_2=0,q_1=1;0\rangle$ state, with no surface plasmon decay or QD dephasing, i.e., $\hslash {\gamma }_s=\hslash {\gamma }_d$ = 0 meV. (a) A case with $g_1<g_2$ corresponding to the initially excited QD1 not being as strongly coupled to the surface plasmon as QD2: $\hslash g_1$ = 12.5 meV, $\hslash g_2=25$ meV. (b) A case with $g_1>g_2$: $\hslash g_1$ = 25 meV, $\hslash g_2$ = 12.5 meV.

Figure 2

Results for a surface plasmon decay width $\hslash {\gamma }_s$ = 100 meV. Upper two panels (a) and (b) show the time dependence of the $|S;s=0\rangle , |A;s=0\rangle , |q_2=0,q_1=0;s=1\rangle$ state populations and concurrence for a two-QD system, initially in the $|q_2=0,q_1=1;0\rangle$ state, and with no QD dephasing, i.e., $\hslash {\gamma }_d$ = 0 meV. The cases with (a) $\hslash g_1$ = 12.5 meV, $\hslash g_2$ = 25 meV and (b) $\hslash g_1$ = 25 meV, $\hslash g_2$ = 12.5 meV are displayed. The lower two panels (c) and (d) are the maximum concurrences found as a function of the QD/plasmon coupling factors, $g_1$ and $g_2$. (c) Corresponds to no QD dephasing, ${\gamma }_d$ = 0, and contains within it the concurrence maxima from the particular cases (a) and (b) above. (d) The corresponding maximum concurrence when the QD dephasing is set to $\hslash {\gamma }_d$ = 2.0 meV. The dashed lines in (c) and (d) represent $g_2$ = $\sqrt{3}{g}_1$.

Figure 3

Asymptotic figure of merit, Eq. (20) for a three-QD system, with one QD initially excited, ${\gamma }_s>0$, and ${\gamma }_d=0$, as a function of the ratios of the QD/plasmon coupling parameters.

Figure 4

(a) and (b) The time dependence of the populations of the $|A\rangle$ and $|S\rangle$ states (traced over all $|s\rangle$) and concurrence for pulsed excitation of an initially cold two-QD/surface plasmon system with parameters $\hslash g_1$ = 12.8 meV, $\hslash g_2$ = 24.9 meV, $F$ = 263.4 $\mathrm{nJ}/{\mathrm{cm}}^2, \tau$ = 12.5 fs, $\hslash {\gamma }_s$ = 186 meV, and $\hslash {\gamma }_d$ = 0 meV. These parameters are the result of a local optimization run. (c) and (d) These same parameter values except for the QD dephasing, which is either (c) $\hslash {\gamma }_d$ = 0.2 meV or (d) $\hslash {\gamma }_d$ = 2 meV.

Figure 5

Maximum concurrence for a parameter sweep of the two-QD system, with $\hslash g_2$ = 30 meV, $\tau$ = 20 fs, $\hslash {\gamma }_s$ = 150 meV, $\hslash {\gamma }_d=2$ meV. The dashed lines represent coupling ratios obeying Eq. (19).

Figure 6

Populations and concurrences for the final parameters from a local optimization run on the three-QD system with $\hslash {\gamma }_d$ fixed at 0.2 meV. The final parameters are $\hslash g_1$ = 14.6 meV, $\hslash g_2=19.3$ meV, $\hslash g_3=19.3$ meV, $F$ = 587.0 $\mathrm{nJ}/{\mathrm{cm}}^2, \tau$ = 36.4 fs, and $\hslash {\gamma }_s=180.4$ meV. (a) Shows the various bipartite concurrences, and (b) shows the QD excitation probabilities. Because $g_2=g_3$, the QD3:QD1 and QD2:QD1 concurrences are identical, as are the QD2 and QD3 excitation probabilities. (c) Shows the time-dependent probabilities of the $|S\rangle$ and $|A\rangle$ states associated with either the QD3:QD1 or QD2:QD1 subsystems and the pulse envelope.

Figure 7

Populations and concurrences for the final parameters from a second local optimization run on the three-QD system with $\hslash {\gamma }_d$ fixed at 0.2 meV. The final parameters are $\hslash g_1$ = 19.0 meV, $\hslash g_2$ = 16.3 meV, $\hslash g_3$ = 21.7 meV, $F$ = 603.4 $\mathrm{nJ}/{\mathrm{cm}}^2, \tau$ = 39.4 fs, and $\hslash {\gamma }_s$ = 107.7 meV. (a) Shows the various bipartite concurrences, and (b) shows the QD excitation probabilities. The pulse envelope and the populations of the $|S\rangle$ and $|A\rangle$ states are shown for the QD3:QD1 pair in (c) and the QD2:QD1 pair in (d).